Chapter 2

Differential Calculus of Functions ofOne Variable

2.1 Functions and Limits

1. Each of the following conditions fails to define a function on any domain. State why.

a. sin f (x) = x

Solution

• If |x| > 1 this equation has no (real) solution for f (x).• If |x| ≤ 1, each of the values

f (x) = arcsinx+ 2nπ, n ∈ Z and f (x) = π − arcsinx+ 2nπ, n ∈ Z

will satisfy the given equation equation. Hence the assigned value f (x) is notunique and therefore f is not a function.

b. ef(x) = − |x|SolutionSince the exponential function g (x) = ex is strictly positive, the given equation has nosolution for any value of x.

c. 1 + x2 + [f (x)]2 = 0

SolutionSince 1 + x2 + [f (x)]2 ≥ 1, the given equation has no solution for any value of x.

d. f (x) [f (x)− 1] = x2

Solution

11

12 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

This equation is equivalent to f2 (x)− f (x)− x2 = 0. Using the quadratic formula, weobtain

f (x) =1±√1 + 4x2

2

Again, the assigned value f (x) is not unique and therefore f is not a function.

3. Find Df .

a. f (x) = tanxSolutionDf = R−

©π2 + nπ | n ∈ Z

ªd. f (x) = sinx

x

SolutionDf = R− {0}

e. e[f(x)]2= x, f (x) ≥ 0

SolutionSince ey

2attains all values greater than or equal to one and none less than one, the

given equation has a solution if and only if x ≥ 1. Hence Df = [1,∞). (of course thisis commensurate with the equivalent equation f (x) =

√lnx)

4. Find limx→x0 f (x) and justify your answers with an -δ proof.

a. limx→1¡x2 + 2x+ 1

¢= 4

SolutionLet f (x) = x2 + 2x+ 1, L = 4, and > 0. Consider

|f (x)− L| =¯̄¡x2 + 2x+ 1

¢− 4¯̄=¯̄x2 + 2x− 3

¯̄= |x+ 3| |x− 1|

Let δ ≤ 1. Then for all x with 0 < |x− 1| < δ ≤ 1

−1 ≤ x− 1 ≤ 1, so 3 ≤ x+ 3 ≤ 5

Hence,|f (x)− L| = |x+ 3| |x− 1| ≤ 5 |x− 1|

Choose δ = min©1, 5

ª, then for all x with 0 < |x− 1| < δ

|f (x)− L| ≤ 5 |x− 1| < 5 ·5=

This completes the proof.

2.1. FUNCTIONS AND LIMITS 13

b. limx→2x3−8x−2 = limx→2

(x−2)(x2+2x+4)x−2 = limx→2

¡x2 + 2x+ 4

¢= 12

SolutionLet f (x) = x3−8

x−2 , L = 12, and > 0. For x 6= 2

|f (x)− L| =¯̄̄̄x3 − 8x− 2 − 12

¯̄̄̄=¯̄x2 + 2x− 8

¯̄= |x+ 4| |x− 2|

Let δ ≤ 1. Then for all x with 0 < |x− 2| < δ ≤ 1

−1 ≤ x− 2 ≤ 1, so 5 ≤ x+ 4 ≤ 7

Hence,|f (x)− L| = |x+ 4| |x− 2| ≤ 7 |x− 2|

Choose δ = min©1, 7

ª, then for all x with 0 < |x− 2| < δ

|f (x)− L| ≤ 7 |x− 2| < 7 ·7=

This completes the proof.

d. limx→4√x = 2

SolutionLet f (x) =

√x,L = 2, and > 0. Consider

|f (x)− L| =¯̄√

x− 2¯̄=

¯̄̄̄x− 4√x+ 2

¯̄̄̄=

1√x+ 2

|x− 4| ≤ 12|x− 4|

Choose δ = 2 , then for all x ∈ Df with 0 < |x− 4| < δ

|f (x)− L| ≤ 12|x− 4| < 1

2· 2 =

This completes the proof.

7. Find limx→x0− f (x) and limx→x0+ f (x), if they exist. Use -δ proofs, where applicable, tojustify your answers.

b. x cos 1x + sin1x + sin

1|x| , x0 = 0

Solution

14 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

• limx→0−³x cos 1x + sin

1x + sin

1|x|

´Observe that

limx→0−

µx cos

1

x+ sin

1

x+ sin

1

|x|

¶= lim

x→0−

µx cos

1

x+ sin

1

x+ sin

µ−1x

¶¶= lim

x→0−

µx cos

1

x+ sin

1

x− sin 1

x

¶= lim

x→0−x cos

1

x

Finally, we will now show that limx→0− x cos1x = 0. Let > 0. Consider¯̄̄̄

x cos1

x− 0¯̄̄̄= |x|

¯̄̄̄cos

1

x

¯̄̄̄≤ |x|

Choose δ = , then for all x with −δ < x < 0¯̄̄̄x cos

1

x− 0¯̄̄̄≤ |x| < δ =

We conclude that

limx→0−

µx cos

1

x+ sin

1

x+ sin

1

|x|

¶= lim

x→0−x cos

1

x= 0

• limx→0+³x cos 1x + sin

1x + sin

1|x|

´Above we proved that limx→0− x cos

1x = 0. In a similar manner it can be shown

that limx→0+ x cos1x = 0, therefore

limx→0+

µx cos

1

x+ sin

1

x+ sin

1

|x|

¶= lim

x→0+

µx cos

1

x+ sin

1

x+ sin

1

x

¶= lim

x→0+

µx cos

1

x+ 2 sin

1

x

¶= lim

x→0+x cos

1

x+ lim

x→0+2 sin

1

x

= limx→0+

2 sin1

x

We now prove that limx→0+ 2 sin1x does not exist. Let L ∈ R, δ > 0 and 0 =

max {|2− L| , |2 + L|}. By the Archimedean property of R, ∃n ∈ N such that

0 < x1 =1

π2 + 2nπ

< δ and 0 < x2 =1

3π2 + 2nπ

< δ

2.1. FUNCTIONS AND LIMITS 15

Note that¯̄̄2 sin 1

x1− L

¯̄̄= |2− L| and

¯̄̄2 sin 1

x2− L

¯̄̄= |−2− L| = |2 + L|. Hence,

for every L ∈ R, there exists an 0 > 0 such that for every δ > 0, there exists an xwith 0 < x < δ, with

¯̄2 sin 1x − L

¯̄≥ 0. This shows that limx→0+ 2 sin

1x does not

exist. Therefore

limx→0+

µx cos

1

x+ sin

1

x+ sin

1

|x|

¶does not exist.

c. |x−1|x2+x−2 , x0 = 1

SolutionUse Theorem 2.1.4 adapted for one-sided limits.

• limx→1−|x−1|

x2+x−2 = limx→1−³− x−1

x2+x−2

´= limx→1−

³− 1

x+2

´= −13

• limx→1+|x−1|

x2+x−2 = limx→1+x−1

x2+x−2 = limx→1+1

x+2 =13

8. Prove: If h (x) ≥ 0 for a < x < x0 and limx→x0− h (x) exists, then limx→x0− h (x) ≥ 0.Conclude from this that if f2 (x) ≥ f1 (x) for a < x < x0, then

limx→x0−

f2 (x) ≥ limx→x0−

f1 (x)

if both limits exist.

Solution

In class I suggested you use a proof by contradiction. The essential ingredient of the proofcan be found in the proof of Theorem 2.1.4. Part 4.

Assume limx→x0− h (x) = L < 0. Then ∃δ > 0 such that for all x with x0 − δ < x < x0

|h (x)− L| < |L|2

This implies that

− |L|2

< h (x)− L <|L|2

so

− |L|2

< h (x) + |L| < |L|2

and thus − 3 |L|2

< h (x) < − |L|2

a contradiction with the fact that h (x) ≥ 0 for a < x < x0.

Finally, choosing h (x) = f2 (x)− f1 (x) yields

limx→x0−

(f2 (x)− f1 (x)) ≥ 0 and thus limx→x0−

f2 (x) ≥ limx→x0−

f1 (x)

Provided both limits exist.

16 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

15. Find limx→∞ f (x) if it exists, and justify your answer directly from Definition 2.1.7.

b. sinxxα (α > 0)

SolutionWe will show that limx→∞

sinxxα = 0. Let > 0. Observe that¯̄̄̄sinx

xα− 0¯̄̄̄=|sinx||x|α ≤

1

|x|α

Choose τ = (1Á )1Áα then for all x > τ¯̄̄̄sinx

xα− 0¯̄̄̄≤ 1

|x|α <1

τα=

1³(1Á )1Áα

´α =This completes the proof.

f. e−x2e2x

SolutionWe will show that limx→∞

³e−x

2e2x´= 0. Let > 0. Observe that for x > 4¯̄̄

e−x2e2x − 0

¯̄̄= e−x

2+2x = e−12x2+2xe−

12x2 = e−

12x(x−4)e−

12x2 < e−

12x2

• Note that if > 1, then e−12x2 < . Hence, with τ = 4 and x > τ¯̄̄e−x

2e2x − 0

¯̄̄< e−

12x2 <

• In case 0 < ≤ 1, we quickly solve the equation

e−12x2 =

for x, yielding x =√−2 ln . Choose τ = max

©4,√−2 ln

ªthen for x > τ¯̄̄

e−x2e2x − 0

¯̄̄< e−

12x2 < e−

12τ2 ≤ e−

12(√−2 ln )2 = eln =

22. Find

c. limx→x01

(x−x0)2k, k is a positive integer.

SolutionObserve that in the extended reals

limx→x0−

1

(x− x0)2k=∞ and lim

x→x0+

1

(x− x0)2k=∞

2.2. CONTINUITY 17

so

limx→x0

1

(x− x0)2k=∞

d. limx→x01

(x−x0)2k+1, k is a positive integer.

SolutionObserve that in the extended reals

limx→x0−

1

(x− x0)2k+1

= −∞ and limx→x0+

1

(x− x0)2k+1

=∞

so, even in the extended reals, the undirected limit

limx→x0

1

(x− x0)2k+1

does not exist.

2.2 Continuity

2. Prove that a function f is continuous at x0 if and only if

limx→x0−

f (x) = limx→x0+

f (x) = f (x0)

Solution

• If:Let > 0. Since limx→x0− f (x) = f (x0) there exists a δ1 > 0 such that

|f (x)− f (x0)| < whenever x0 − δ1 < x ≤ x0

Similarly, since limx→x0+ f (x) = f (x0), there exists a δ2 > 0 such that

|f (x)− f (x0)| < whenever x0 ≤ x < x0 + δ2

Choose δ = min {δ1, δ2}, then

|f (x)− f (x0)| < whenever |x− x0| < δ

This shows that f is continuous at x0.

18 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

• Only if:Let > 0. Since f is continuous at x0, there exists a δ > 0 such that

|f (x)− f (x0)| < whenever |x− x0| < δ

This implies that

|f (x)− f (x0)| < whenever x0 − δ < x ≤ x0, and

|f (x)− f (x0)| < whenever x0 ≤ x < x0 + δ

Hence limx→x0− f (x) = f (x0) and limx→x0+ f (x) = f (x0).

3. Determine whether f is continuous from the left or from the right at x0.

c. f (x) = 1x (x0 = 0)

SolutionSince f (0) is undefined, the function f is neither continuous from the left at 0, norcontinuous from the right at 0.

g. f (x) =½

x+|x|(1+x)x sin 1x x 6= 0

1 x = 0(x0 = 0)

SolutionObserve

limx→0−

f (x) = limx→0−

x+ |x| (1 + x)

xsin

1

x= lim

x→0−x− x (1 + x)

xsin

1

x

= limx→0−

µ−x sin 1

x

¶= 0 6= 1 = f (0)

Hence, the function f is not continuous from the left at 0.

Note: The answer in the back of the book is not correct.Similarly

limx→0+

f (x) = limx→0+

x+ |x| (1 + x)

xsin

1

x

= limx→0+

x+ x (1 + x)

xsin

1

x= lim

x→0+(2 + x) sin

1

x

This limit is undefined. So, f is not continuous from the right at 0 either.

2.2. CONTINUITY 19

5. Let

g (x) =

√x

x− 1

On which of the following intervals is f continuous according to definition 2.2.3:

[0, 1) , (0, 1) , (0, 1] , [1,∞) , (1,∞)?Solution

[0, 1) , (0, 1) , and (1,∞).

11. Prove that the function g (x) = log x is continuous on (0,∞). Take the following propertiesas given.

(a) limx→1 g (x) = 0

(b) g (x1) + g (x2) = g (x1x2) if x1, x2 > 0.

Solution

Let > 0 and x0 ∈ (0,∞). In class we showed that property (b) is equivalent to

g (x1)− g (x2) = g

µx1x2

¶if x1, x2 > 0

Let x ∈ (0,∞). Consider

|g (x)− g (x0)| =¯̄̄̄g

µx

x0

¶¯̄̄̄Since limx→1 g (x) = 0, there exists a δ1 > 0, such that

|g (u)| = |g (u)− 0| = |g (u)− g (1)| < whenever |u− 1| < δ1

Observe that ¯̄̄̄x

x0− 1¯̄̄̄=

¯̄̄̄x− x0x0

¯̄̄̄< δ1 whenever |x− x0| < δ1 |x0|

Choose δ = δ1 |x0| then, by letting xx0play the role of u, we may conclude that

|g (x)− g (x0)| =¯̄̄̄g

µx

x0

¶¯̄̄̄< whenever |x− x0| < δ

Therefore g is continuous at x0, and since x0 was chosen arbitrarily on (0,∞) this shows thatg is continuous on (0,∞).

20 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

16. Let |f | be the function whose value at each x in Df is |f (x)|. Prove: If f is continuous atx0, then so is |f |. Is the converse true?Solution

Let > 0, then

||f | (x)− |f | (x0)| = ||f (x)|− |f (x0)|| ≤ |f (x)− f (x0)|

Moreover, since f is continuous at x0, there exists a δ > 0 such that |f (x)− f (x0)| <whenever |x− x0| < δ, so

||f | (x)− |f | (x0)| ≤ |f (x)− f (x0)| < whenever |x− x0| < δ

Hence, the function |f | is continuous at x0.The converse is not true. Consider for instance the function

f (x) =

½ |x|x if x 6= 01 if x = 0

Then |f | is continuous at 0, but f is not.

20. (a) Let f1 and f2 be continuous at x0 and define

F (x) = max {f1 (x) , f2 (x)}

Show that F is continuous at x0.SolutionThe key idea for this proof is to make a distinction between the case that f1 (x0) =f2 (x0), and the case that f1 (x0) 6= f2 (x0). Let > 0.

• Case 1: f1 (x0) = f2 (x0)Note that in this case F (x0) = max {f1 (x0) , f2 (x0)} = f1 (x0) = f2 (x0). Since f1is continuous at x0, there exists a δ1 such that

|f1 (x)− F (x0)| = |f1 (x)− f1 (x0)| < whenever |x− x0| < δ1

Similarly, since f2 is continuous at x0, there exists a δ2 such that

|f2 (x)− F (x0)| = |f2 (x)− f2 (x0)| < whenever |x− x0| < δ2

Choose δ = min {δ1, δ2}. Then, because F (x) either equals either f1 (x) or f2 (x),

|F (x)− F (x0)| < whenever |x− x0| < δ

This shows that F is continuous at x0.

2.2. CONTINUITY 21

• Case 2: f1 (x0) 6= f2 (x0)Since f1 is continuous at x0, there exists a δ1 such that

|f1 (x)− f1 (x0)| < whenever |x− x0| < δ1

Similarly, since f2 is continuous at x0, there exists a δ2 such that

|f2 (x)− f2 (x0)| < whenever |x− x0| < δ2

Choose δ = min {δ1, δ2}. Without loss of generality, we may assume that f1 (x0) >f2 (x0). Take < f1(x0)−f2(x0)

2 . This assures that whenever |x− x0| < δ

|F (x)− F (x0)| = |f1 (x)− f1 (x0)| <

Again, this shows that F is continuous at x0.

(b) Let f1, f2, . . . , fn be continuous at x0 and define

F (x) = max {f1 (x) , f2 (x) , . . . , fn (x)}

Show that F is continuous at x0.SolutionUse mathematical induction. Let Pn denote the proposition mentioned above. Part(a) of this exercise shows that P2 is true. Let n denote any positive integer greater thanor equal to 2 and assume that P2, P3, . . . Pn are all true. Let

h (x) = max {f1 (x) , f2 (x) , . . . , fn (x)}

then h and fn+1 are both continuous at x0 and therefore

F (x) = max {f1 (x) , f2 (x) , . . . , fn+1 (x)} = max {h (x) , fn+1 (x)}

is continuous at x0. Hence Pn+1 is true and by the principle of mathematical inductionwe may conclude that Pn is true for all positive integers greater than or equal to 2.

21. Find the domains of f ◦ g and g ◦ f .

a. f (x) =√x, g (x) = 1− x2

SolutionDf = [0,∞) and Dg = (−∞,∞).• Let T = [−1, 1]. Then T ⊂ Dg and g (x) ∈ Df whenever x ∈ T . The set T is thedomain of f ◦ g.

22 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

• Let T = [0,∞). Then T ⊂ Df and f (x) ∈ Dg whenever x ∈ T . The set T is thedomain of g ◦ f .

c. f (x) = 11−x2 , g (x) = cosx

SolutionDf = {x | x 6= −1, 1} and Dg = (−∞,∞).• Let T = {x | x 6= nπ, n ∈ Z}. Then T ⊂ Dg and g (x) ∈ Df whenever x ∈ T . Theset T is the domain of f ◦ g.

• Let T = {x | x 6= −1, 1}. Then T ⊂ Df and f (x) ∈ Dg whenever x ∈ T . The setT is the domain of g ◦ f .

23. Use Theorem 2.2.7 to find all points x0 at which the following functions are continuous.

a.√1− x2

SolutionLet f (x) =

√x and g (x) = 1 − x2. Then g is continuous at all real x0, while f is

continuous at all x0 > 0. We conclude that f ◦ g is continuous at all x0 with 1−x20 > 0;that is the set ©

x0 | 1− x20 > 0ª= (−1, 1)

g.¡1− sin2 x

¢−1Á2SolutionLet f (x) = 1√

xand g (x) = 1 − sin2 x = cos2 x. Then g is continuous at all real x0,

while f is continuous at all x0 > 0. We conclude that f ◦ g is continuous at all x0 withcos2 x0 > 0; that is the set©

x0 | cos2 x0 > 0ª=©x0 | cos2 x0 6= 0

ª=nx0 | x0 6=

π

2+ kπ, k ∈ Z

oNote: The answer in the back of the book is not correct.

24. Complete the proof of Theorem 2.2.9 by showing that there is an x2 ∈ [a, b] such thatf (x2) = β.

Solution

Recall that f is continuous on [a, b] and β = supx∈[a,b] f (x) = sup {f (x) | x ∈ [a, b]}. Supposethere is no x2 ∈ [a, b] such that f (x2) = β. Then f (x) < β for all x ∈ [a, b]. Let t ∈ [a, b],then f (t) < β, so

f (t) <f (t) + β

2< β

2.3. DIFFERENTIABLE FUNCTIONS OF ONE VARIABLE 23

Moreover, since f is continuous at t, there exists an open interval It containing t, such that

f (x) <f (t) + β

2for all x ∈ It ∩ [a, b]

Note that H = {It | t ∈ [a, b]} is an open covering of [a, b] , and since [a, b] is compact, H canbe reduced to a finite sub-cover, say

{Iti | 1 ≤ i ≤ n}

Let

β1 = max1≤i≤n

f (ti) + β

2

Observe that β1 < β, andf (x) < β1 for all x ∈ [a, b]

This implies that β1 is an upper bound for the set V = {f (x) | x ∈ [a, b]} which is lessthan β = supV , a contradiction. We conclude that there must be an x2 ∈ [a, b] such thatf (x2) = β.

2.3 Differentiable Functions of One Variable

3. Use Lemma 2.3.2. to prove that if f 0 (x0) > 0, there is a δ > 0 such that

f (x) < f (x0) if x0 − δ < x < x0 and f (x) > f (x0) if x0 < x < x0 + δ

Solution

Since limx→x0 E (x) = 0, there exists a δ > 0 such that |E (x)| < f 0 (x0) whenever 0 <|x− x0| < δ. Therefore, if 0 < |x− x0| < δ,

f 0 (x0) +E (x) > 0

Moreover, sincef (x)− f (x0) =

£f 0 (x0) +E (x)

¤(x− x0)

the desired result follows.

5. Find all derivatives of f (x) = xn−1 |x|, where n is a positive integer.Solution

Recall thatd

dx|x| =

½ |x|x if x 6= 0undefined if x = 0

24 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

• Using the product rule we find that if x 6= 0

— f 0 (x) = (n− 1)xn−2 |x| + xn−1 |x|x = (n− 1)xn−2 |x| + xn−2 |x| = nxn−2 |x|, andsimilarly

— f 00 (x) = n (n− 1)xn−3 |x|· · ·

— f (k) (x) = n (n− 1) . . . (n− k + 1)xn−k−1 |x| for 1 ≤ k ≤ n− 1Observe that the result for f (k) (x) is based on the premise that f (k−1) (x) is of theform axb |x| where a and b are positive integers. The (n− 1)stderivative of f is nolonger of this form

f (n−1) (x) = n (n− 1) . . . 2 |x| = n! |x|

Hence, its derivative needs to be examined separately

— f (n) (x) =

½−n! if x < 0n! if x > 0

, and

— f (k) (x) = 0 if k > n.

• Next we examine f (k) (0). Observe that for integers m

µd

dxxm |x|

¶¯̄̄̄x=0

=

(limx→0

xm|x|−0x−0 = limx→0 x(m−1) |x| = 0 if m ≥ 1

undefined if m < 1

• Combining the results for x = 0 and x 6= 0 we obtain

— f (k) (x) = n (n− 1) . . . (n− k + 1)xn−k−1 |x| for 1 ≤ k ≤ n− 1Note: The answer in the back of the book is not correct.

— f (n) (x) =

⎧⎨⎩−n! if x < 0undefined if x = 0n! if x > 0

— f (k) (x) =

½0 if x 6= 0undefined if x = 0

for k ≥ n+ 1.

10. Prove Theorem 2.3.4 (b).

If f and g are differentiable at x0, then so is f − g and

(f − g)0 (x0) = f 0 (x0)− g0 (x0)

Solution

2.3. DIFFERENTIABLE FUNCTIONS OF ONE VARIABLE 25

Observe

limx→x0

(f − g) (x)− (f − g) (x0)

x− x0= lim

x→x0

f (x)− g (x)− [f (x0)− g (x0)]

x− x0

= limx→x0

∙f (x)− f (x0)− [g (x)− g (x0)]

x− x0

¸= lim

x→x0

f (x)− f (x0)

x− x0− lim

x→x0

g (x)− g (x0)

x− x0= f 0 (x0)− g0 (x0)

We conclude that (f − g)0 (x0) exists and equals f 0 (x0)− g0 (x0).

11. Prove Theorem 2.3.4 (d).

If f and g are differentiable at x0 and g (x0) 6= 0, then f/g is differentiable at x0 andµf

g

¶0(x0) =

f 0 (x0) g (x0)− f (x0) g0 (x0)

[g (x0)]2

Solution

Observe

limx→x0

³fg

´(x)−

³fg

´(x0)

x− x0= lim

x→x0

f(x)g(x) −

f(x0)g(x0)

x− x0= lim

x→x0

f (x) g (x0)− f (x0) g (x)

(x− x0) g (x) g (x0)

= limx→x0

f (x) g (x0)− f (x0) g (x0) + f (x0) g (x0)− f (x0) g (x)

(x− x0) g (x) g (x0)

= limx→x0

[f (x)− f (x0)] g (x0)− f (x0) [g (x)− g (x0)]

(x− x0) g (x) g (x0)

= limx→x0

f(x)−f(x0)x−x0 g (x0)− f (x0)

g(x)−g(x0)x−x0

g (x) g (x0)

=

hlimx→x0

f(x)−f(x0)x−x0

ig (x0)− f (x0)

hlimx→x0

g(x)−g(x0)x−x0

i[limx→x0 g (x)] g (x0)

Since g is differentiable at x0, g is continuous at x0, so limx→x0 g (x) = g (x0), therefore

limx→x0

³fg

´(x)−

³fg

´(x0)

x− x0=

f 0 (x0) g (x0)− f (x0) g0 (x0)

[g (x0)]2

which proves the stated result.

26 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

15. a. Show that f 0+ (a) = f 0 (a+) if both quantities exist.SolutionIn class I suggested you use the Mean Value Theorem. We will do just that. If f 0 (a+)exists, then f 0 exists on some open interval (a, a+ δ). If the right hand derivative f 0+ (a)exists, then f is continuous from the right at a. Let x∗ ∈ (a, a+ δ), then f is continuouson [a, x∗] and differentiable on (a, x∗), so the Mean Value Theorem applies and thereexists a c ∈ (a, x∗) such that

f (x∗)− f (a)

x∗ − a= f 0 (c)

This implies

f 0+ (a) = limx∗→a+

f (x∗)− f (a)

x∗ − a= lim

x∗→a+f 0 (c)

Finally, note that because c ∈ (a, x∗), c→ a+ if x∗ → a+ and since limx→a+ f 0 (x) exists

limx∗→a+

f 0 (c) = limx→a+

f 0 (x) = f 0 (a+)

We conclude that f 0+ (a) = f 0 (a+).

b. Example 2.3.4 shows that f 0+ (a) may exist even if f0 (a+) does not. Give an example

where f 0 (a+) exists but f 0+ (a) does not.SolutionTake a simple function which is not continuous from the right at a. For instance

f (x) =

½1 if x = 0x if x > 0

then

limx→0+

f (x)− f (0)

x− 0 = limx→0+

x− 1x

is undefined

while f 0 (0+) = 1.

c. Complete the following statement so it becomes a theorem, and prove the theorem: ”Iff 0 (a+) exists and f is ____ at a, then f 0+ (a) = f 0 (a+).”Solution”If f 0 (a+) exists and f is continuous from the right at a, then f 0+ (a) = f 0 (a+).”ProofIf f 0 (a+) exists, then f 0 exists on some open interval (a, a+ δ). Let x∗ ∈ (a, a+ δ),then since f is continuous from the right at a, f is continuous on [a, x∗] and differentiableon (a, x∗), so the Mean Value Theorem applies and there exists a c ∈ (a, x∗) such that

f (x∗)− f (a)

x∗ − a= f 0 (c)

2.3. DIFFERENTIABLE FUNCTIONS OF ONE VARIABLE 27

This implies

limx∗→a+

f (x∗)− f (a)

x∗ − a= lim

x∗→a+f 0 (c)

Note that because c ∈ (a, x∗), c→ a+ if x∗ → a+ and since limx→a+ f 0 (x) exists

limx∗→a+

f 0 (c) = limx→a+

f 0 (x) = f 0 (a+)

We conclude that f 0+ (a) exists and equals f0 (a+).

20. Let n be a positive integer and

f (x) =sinnx

n sinx, x 6= kπ (k is integer) .

a. Define f (kπ) such that f is continuous at kπ.SolutionLet

f (kπ) = limx→kπ

f (x) = limx→kπ

sinnx

n sinx= lim

x→kπ

n cosnx

n cosx

= limx→kπ

cosnx

cosx=cosnkπ

cos kπ=(−1)nk

(−1)k= (−1)(n−1)k

and redefine f as

f (x) =

½ sinnxn sinx x 6= kπ, k ∈ Z(−1)(n−1)k x = kπ, k ∈ Z

b. Show that if x is a local extreme point of f , then

|f (x)| =£1 +

¡n2 − 1

¢sin2 x

¤−1/2HINT: Express sinnx and cosnx in terms of f (x) and f 0 (x), and add their squares toobtain a useful identity.SolutionFirst consider the case that x 6= kπ, k ∈ Z, then f 0 (x) = 0. Observe that

sinnx = nf (x) sinx

and

0 = f 0 (x) =n cosnx

n sinx− sinnx cosx

n sin2 x

=cosnx

sinx− f (x)

cosx

sinx

28 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

hencecosnx = f (x) cosx

therefore

1 = cos2 nx+ sin2 nx = [f (x) cosx]2 + [nf (x) sinx]2

= f2 (x)£cos2 x+ n2 sin2 x

¤= f2 (x)

£1 +

¡n2 − 1

¢sin2 x

¤which implies that

|f (x)| =£1 +

¡n2 − 1

¢sin2 x

¤−1/2Moreover, if k ∈ Z, |f (kπ)| =

¯̄̄(−1)(n−1)k

¯̄̄= 1, so the formula

|f (x)| =£1 +

¡n2 − 1

¢sin2 x

¤−1/2is true even if x = kπ, k ∈ Z.

c. Show that |f (x)| ≤ 1 for all x. For what values of x is equality attained?SolutionFor integer k, let I denote the closed and bounded interval [(k − 1)π, kπ]. Then f iscontinuous on I and by the Extreme Value Theorem f must have a minimum m anda maximum M on I. Therefore there exists a local extreme point x on I such thatm = f (x). Hence, by the result of Part (b) |m| = |f (x)| ≤ 1. In a similar fashion wecan prove that |M | ≤ 1. This means that for all x in I

−1 ≤ − |m| ≤ m ≤ f (x) ≤M ≤ |M | ≤ 1

so|f (x)| ≤ 1 for all x in I.

Because k was an arbitrary integer this implies that |f (x)| ≤ 1 for all real x. If n = 1equality is attained for all x ∈ R, and if n > 1 equality is attained for x = kπ, k ∈ Z.

2.4 L’Hospital’s Rule

In Exercises 2 - 40, find the indicated limits.

6. limx→0log(1+x)

x

Solution

limx→0

log (1 + x)

x= lim

x→0

11+x

1= 1

2.5. TAYLOR’S THEOREM 29

7. limx→∞ ex sin e−x2

Solution

limx→∞

ex sin e−x2= lim

x→∞sin e−x

2

e−x= lim

x→∞−2xe−x2 cos e−x2

−e−x

= limx→∞

2x

ex2−x= lim

x→∞2

(2x− 1) ex2−x

= limx→∞

2

(2x− 1) ex(x−1)= 0

20. limx→0 (1 + x)1/x

Solution

limx→0

(1 + x)1/x = elimx→0log(1+x)

x = elimx→0

11+x1 = e1 = e

23. limx→0+ xα log x

Solution

Observe that if α ≤ 0, limx→0+ xα log x = −∞. When α > 0, then

limx→0+

xα log x = limx→0+

log x

x−α= lim

x→0+

1x

−αx−α−1 = limx→0+

− 1αxα = 0

26. limx→1+³x+1x−1

´√x2−1Solution

limx→1+

µx+ 1

x− 1

¶√x2−1= elimx→1+[

√x2−1 log(x+1x−1)]

= elimx→1+[√x+1

√x−1(log(x+1)−log(x−1))]

= elimx→1+[√x+1

√x−1 log(x+1)−

√x+1

√x−1 log(x−1)]

= elimx→1+[−√2√x−1 log(x−1)] = e

limx→1+

∙−√2

log(x−1)(x−1)−1/2

¸

= elimx→1+

∙2√2

1x−1

(x−1)−3/2

¸= elimx→1+[2

√2√x−1] = 1

2.5 Taylor’s Theorem

2. Suppose f (n+1) (x0) exists, and let Tn be the nth Taylor polynomial of f about x0. Showthat the function

En (x) =

(f(x)−Tn(x)(x−x0)n if x ∈ Df − {x0}0 if x = 0

30 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

Is differentiable at x0 and find E0n (x0).

Solution

Observe

limx→x0

En (x)−En (x0)

x− x0= lim

x→x0

En (x)− 0x− x0

= limx→x0

f(x)−Tn(x)(x−x0)n

x− x0= lim

x→x0

f (x)− Tn (x)

(x− x0)n+1

Let Tn+1 (x) denote the (n+ 1)st Taylor polynomial of f about x0. Then

limx→x0

En (x)−En (x0)

x− x0= lim

x→x0

f (x)− Tn (x)

(x− x0)n+1

= limx→x0

f (x)− Tn+1 (x)

(x− x0)n+1 + lim

x→x0

Tn+1 (x)− Tn (x)

(x− x0)n+1

= 0 + limx→x0

"1

(x− x0)n+1

(f (n+1) (x0)

(n+ 1)!(x− x0)

n+1

)#

=f (n+1) (x0)

(n+ 1)!

We conclude that

E0n (x0) =f (n+1) (x0)

(n+ 1)!

4. a. Prove: if f 00 (x0) exists, then

limh→0

f (x0 + h)− 2f (x0) + f (x0 − h)

h2= f 00 (x0)

SolutionSince f 00 (x0) exists, the second Taylor polynomial T2 of f about x0 is defined. We willcompare f (x0 + h) to T2 (x0 + h), and f (x0 + h) to T2 (x0 + h). Recall

limx→x0

f (x)− T2 (x)

(x− x0)2 = 0

If in this result x is replaced by x0 + h we obtain

limh→0

f (x0 + h)− T2 (x0 + h)

h2= 0

2.5. TAYLOR’S THEOREM 31

and similarly

limh→0

f (x0 − h)− T2 (x0 − h)

h2= 0

Now observe

limh→0

f (x0 + h)− 2f (x0) + f (x0 − h)

h2

= limh→0

f (x0 + h)− T2 (x0 + h)

h2+ lim

h→0

T2 (x0 + h)− 2f (x0) + T2 (x0 − h)

h2

+ limh→0

f (x0 − h)− T2 (x0 − h)

h2

= limh→0

T2 (x0 + h)− 2f (x0) + T2 (x0 − h)

h2

= limh→0

∙f (x0) + f 0 (x0)h+ 1/2f 00 (x0)h2

h2− 2f (x0)

h2

+f (x0)− f 0 (x0)h+ 1/2f 00 (x0)h2

h2

¸= lim

h→0

f 00 (x0)h2

h2= lim

h→0f 00 (x0) = f 00 (x0)

b. Prove or give a counter example: If the limit in Part a exists, then so does f 00 (x0) andthey are equal.SolutionNotice that the proof of Part a is based on the assumption that f 00 (x0) exists. Withoutthat assumption T2 is undefined and the proof falls apart. To generate a counterexamplefor the given statement we will look for a function f that quadratically approaches f (x0)as x approaches x0 and for which f 00 (x0) is undefined. With x0 = 0, the functionf (x) = x |x| satisfies those requirements. We now verify that this function truly is acounterexample.

• f 0 (0) = limx→0x|x|−0

x = limx→0 |x| = 0, and if x 6= 0, f 0 (x) = |x|+ x|x|x = 2 |x|. So

limx→0

f 0 (x)− f 0 (0)

x= lim

x→02 |x|x

is undefined

Therefore f 00 (0) is undefined.• Observe

limh→0

f (h)− 2f (0) + f (−h)h2

= limh→0

h |h|− h |−h|h2

= limh→0

0

h2= lim

h→00 = 0

32 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

This means that with x0 = 0, limh→0f(x0+h)−2f(x0)+f(x0−h)

h2 = 0.

8. a. Let

h (x) =nX

r=0

αr (x− x0)r

be a polynomial of degree ≤ n such that

limx→x0

h (x)

(x− x0)n = 0

Show that αr = 0 for 0 ≤ r ≤ n.SolutionWe use induction on n. Let Pn denote the proposition above.

• First we verify that P0 is true. Note that if n = 0, them h (x) = α0 and (x− x0)n =

1, solimx→x0

α0 = 0

Hence, α0 = 0.• Let k denote a nonnegative integer and let Pk be true. Additionally let

h (x) =k+1Xr=0

αr (x− x0)r and lim

x→x0

h (x)

(x− x0)k+1

= 0

Then

limx→x0

h (x) = limx→x0

"h (x)

(x− x0)k+1

(x− x0)k+1

#= 0 · 0 = 0

and since h is a polynomial it is continuous everywhere, so

h (x0) = limx→x0

h (x) = 0

which means α0 = 0. Thus, with m = r − 1

h (x) =k+1Xr=1

αr (x− x0)r =

kXm=0

αm+1 (x− x0)m+1

and

0 = limx→x0

h (x)

(x− x0)k+1

= limx→x0

Pkm=0 αm+1 (x− x0)

m+1

(x− x0)k+1

= limx→x0

Pkm=0 αm+1 (x− x0)

m

(x− x0)k

2.5. TAYLOR’S THEOREM 33

Observe that the numerator in the last expression is a polynomial of degree ≤ k, sothe induction assumption applies and we may conclude that in addition to the factthat α0 = 0, also αr = 0 for 1 ≤ r ≤ k + 1. Hence Pk+1 is true. This completesthe proof.

b. Suppose f is n times differentiable at x0 and p =Pn

r=0 αr (x− x0)r is a polynomial of

degree ≤ n such that

limx→x0

f (x)− p (x)

(x− x0)n = 0

Show that

αr =f (r) (x0)

r!if 0 ≤ r ≤ n

that is, p = Tn. the nth Taylor polynomial of f about x0.SolutionObserve

0 = limx→x0

f (x)− p (x)

(x− x0)n = lim

x→x0

f (x)− Tn (x)

(x− x0)n + lim

x→x0

Tn (x)− p (x)

(x− x0)n

= 0 + limx→x0

Tn (x)− p (x)

(x− x0)n

we conclude that

limx→x0

Tn (x)− p (x)

(x− x0)n = 0

and by Part a, this implies that Tn (x)− p (x) is identically equal to zero, so p = Tn.

16. Find an upper bound for the magnitude of the error in the approximation.

b.√1 + x ≈ 1 + x

2 , |x| <18

SolutionLet f (x) =

√1 + x, then use Taylor’s theorem with n = 1 and x0 = 0.

√1 + x−

h1 +

x

2

i=

f (2) (c)

2!x2

so ¯̄̄√1 + x−

h1 +

x

2

i¯̄̄=

¯̄̄̄¯f (2) (c)2!

x2

¯̄̄̄¯ ≤ 12

µ1

8

¶2 ¯̄̄f (2) (c)

¯̄̄=

1

128

¯̄̄f (2) (c)

¯̄̄Next we estimate

¯̄f (2) (c)

¯̄= 1

4(c+1)3/2for |c| < 1

8 . Observe¯̄̄f (2) (c)

¯̄̄=

1

4 (c+ 1)3/2≤ 1

4¡−18 + 1

¢3/2 = 4

49

√14

34 CHAPTER 2. DIFFERENTIAL CALCULUS OF FUNCTIONS OF ONE VARIABLE

Hence, an upper bound for the magnitude of the error is given by

1

128· 449

√14 =

1

1568

√14 ≈ 2. 386 3× 10−3

Note: The answer in the back of the book is not correct.

d. log x ≈ (x− 1)− (x−1)22 + (x−1)3

3 , |x− 1| < 164

SolutionLet f (x) = log x, then use Taylor’s theorem with n = 3 and x0 = 1.

log x−"(x− 1)− (x− 1)

2

2+(x− 1)3

3

#=

f (4) (c)

4!(x− 1)4

so ¯̄̄̄¯log x−

"(x− 1)− (x− 1)

2

2+(x− 1)3

3

#¯̄̄̄¯

=

¯̄̄̄¯f (4) (c)4!

(x− 1)4¯̄̄̄¯ ≤ 1

24

µ1

64

¶4 ¯̄̄f (4) (c)

¯̄̄=

1

402 653 184

¯̄̄f (4) (c)

¯̄̄Next we estimate

¯̄f (4) (c)

¯̄= 6

c4 for |c− 1| <164 . Observe¯̄̄

f (4) (c)¯̄̄=6

c4≤ 6¡

1− 164

¢4 = 33 554 432

5250 987

Hence, an upper bound for the magnitude of the error is given by

1

402 653 184· 33 554 4325250 987

=1

63 011 844≈ 1. 587× 10−8